How to make an IK chain stay above another IK chain
In an IK chain, you can limit bone rotations to keep them within a certain range of movement. Select the base bone in your IK chain and limit one of its rotation values (in this case either X or Z) until the mesh doesn't intersect with the bow's arm. Repeat this process down the chain to get a nice curve running along the arm. When limiting, it is not necessary to set both negative and positive limits to custom values. Most of the time you will want to keep one side completely open (+-180 degrees) while limiting the other. These limit values have hard limits of +-180 degrees because any rotation beyond will simply flip the IK solution to the other side. For more intuitive and predictable visual feedback, I recommend putting your IK chain bones in one of the many Euler rotation modes.